I know it is not in Limassol, but there is a place close to Agios Georgios monastery, just at the sea side. Between Governor's beach and Limassol. It is about 10 minutes drive from Limassol touristic area

There is also wonderful fish tavern in Zygi (20 minutes drive from Limassol) but definitely worth it.

I know the place barryH is recommending - i just love it. The place is called Agios Georgios Almandou, I don't remember the name of the tavern. Going from Limassol you need to take exit for Pentakomo village and the go along Agios Georgios signs. When you see the gate to the church/monastery take the left turn, which goes directly to the sea = fish tavern.